Cytoskeleton-Membrane Interactions in Neuronal Growth Cones: A Finite Analysis Study

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Kathleen B. Allen

F. Mert Sasoglu

Bradley E. Laytone-mail: [email protected]

Department of Mechanical Engineeringand Mechanics,

Drexel University,3141 Chestnut Street,

Philadelphia, PA 19104

Cytoskeleton-MembraneInteractions in Neuronal GrowthCones: A Finite Analysis StudyRevealing the molecular events of neuronal growth is critical to obtaining a deeperunderstanding of nervous system development, neural injury response, and neural tissueengineering. Central to this is the need to understand the mechanical interactions be-tween the cytoskeleton and the cell membrane, and how these interactions affect theoverall growth mechanics of neurons. Using finite element analysis, the stress in themembrane produced by an actin filament or a microtubule acting against a deformablemembrane was modeled, and the deformation, stress, and strain were computed for themembrane. Parameters to represent the flexural rigidities of the well-studied actin andtubulin cytoskeletal proteins, as well as the mechanical properties of cell membranes,were used in the simulations. Our model predicts that a single actin filament is able toproduce a normal contact stress on the cell membrane that is sufficient to cause mem-brane deformation but not growth. Our model also predicts that under clamped boundaryconditions a filament with a buckling strength equal to or smaller than an actin filamentwould not cause the areal strain in the membrane to exceed 3%, and therefore thefilament is incapable of causing membrane rupture or puncture to a safety factor of�15–25. Decreasing the radius of the membrane upon which the normal contact stressis acting allows an increase in the amount of normal contact stress that the membranecan withstand before rupture. The model predicts that a 50 nm radius membrane canwithstand �4 MPa of normal contact stress before membrane rupture whereas a 250 nmradius membrane can withstand �2.5 MPa. Understanding how the mechanical proper-ties of cytoskeletal elements have coevolved with their respective cell membranes mayyield insights into the events that gave rise to the sequences and superquaternary struc-tures of the major cytoskeletal proteins. Additionally, numerical modeling of membranescan be used to analyze the forces and stresses generated by nanoscale biological probesduring cellular injection. �DOI: 10.1115/1.3005337�

Keywords: cytoskeleton-membrane interaction, protein mechanics, finite elementanalysis, cellular injection

ackgroundIn the peripheral nervous system, axons can grow to be a meter

n length, with diameters of only 1–10 �m. Peripheral neurons,nd some neurons within the central nervous system, such ashose of the optic tract, with their enormous aspect ratios, activelyush forth into growing, developing, or healing tissue. Mainte-ance of these million-to-one aspect ratio structures requires in-ernal dynamic protein struts. The primary proteins include micro-ubules, comprised of polymerized tubulin, microfilaments,omprised of polymerized actin, and intermediate filaments, com-rised of polymerized neurofilament light �NFL�, neurofilamentiddle �NFM�, and neurofilament heavy �NFH� subunits. These

ytoskeletal proteins are the most abundant proteins in neurons1�, with actin comprising up to 15% of the total protein in the cell2�. The polymerization and interactions of these proteins driveell growth and maintain the morphology of adult cells �3–6�.

Microtubules are typically regarded as the primary compressivetructural elements. Additionally, they act as force sensors, detect-ng the magnitude and direction of the force causing them to com-ress �3�. Actin networks, which are typically regarded as support-ng tensile forces in cells, also sustain compressive loads in therowth cone region of the axon where an actin cortex is main-

Contributed by the Bioengineering Division of ASME for publication in the JOUR-

AL OF BIOMECHANICAL ENGINEERING. Manuscript received March 6, 2008; final manu-cript received June 16, 2008; published online December 10, 2008. Review con-
ucted by Cheng Zhu.
ournal of Biomechanical Engineering Copyright © 20

om: http://biomechanical.asmedigitalcollection.asme.org/ on 10/04/2013 T

tained �4,5�. These two internal compression-sustaining structuresare responsible for maintaining neuronal membrane tension �6�.Previous work that has focused on determining the flexural rigid-ity of these isolated cytoskeletal proteins includes observation ofthermal fluctuations �7�, hydrodynamic flow �8�, optical tweezers�9�, and microneedle manipulation �10�.

Living cells have been described as having a tensegrity archi-tecture, with their cytoskeletal elements forming a lattice supportstructure that flattens when the cell is adhered to a surface androunds to a spherical shape when the cell is unattached �11�. Ifcells were true tensegrity structures, the protein filaments withinthe cell would always seek a minimal energy confirmation, withall of the protein filaments sustaining equal strain energy densi-ties. Microtubules in the growth cones of neurons and in fibroblastcells have been observed to buckle, bend, or break at the leadingedge of the cell �12–14� indicating that stress is not always evenlydistributed throughout the cytoskeleton. A dynamic balance offorces must therefore exist between the cytoskeleton and the neu-ronal membrane that allows the cytoskeleton to not only supportthe stress of the membrane but also to push the membrane forwardduring development, and then sustain this shape after growth iscomplete. An understanding of this balance is important for deter-mining how nerves respond to injury, e.g., the regrowth of periph-eral nerve cells through scar tissue, in response to various phar-maceuticals, and also for tissue engineering applications.

Recently, Atilgan et al. �15� used an energy-based membrane-cytoskeleton interaction model to predict that the polymerization
energy of a single actin filament is insufficient to initiate the for-
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ation of a filopodium. Rather, two to three are required. Theirodel predicts that at an intracellular concentration of 500 �M,

he amount of free energy reduction caused by actin polymeriza-ion of a single actin filament, is sufficient to overcome the con-omitant rise in free energy caused by the resulting elastic mem-rane deformation. Using models employing uniform stressistributions throughout the cytoskeleton, finite element methodsave been used to determine the strain on the nucleus of an en-othelial cell caused by cytoskeletal deformation during cellounding �16�. Other works on force generation through actin po-ymerization studied the elastic Brownian ratchet model �12�. Inhis model, the elastic forces of growing actin filaments push theell forward. While these models are valuable in predicting theoalescence of nearby actin filaments into filament bundles, ex-laining filopodia growth rates through a thermal ratchet modelnd understanding the mechanism in which bacteria migrate, theyo not consider the stress state within the membrane as it is de-ormed under natural or artificial growth conditions or under simi-ar externally induced cell probing conditions.

This paper thus has a twofold goal. The first is to introduce theoncept that there exists a “natural mechanical safety factor” builtnto the mechanical relationship between the cell membrane andhe cytoskeleton, such that the growth rate and forces of the cy-oskeletal elements do not rupture the cell. The second is to quan-ify the stress and strain state of the membrane as it undergoesontact and penetration by nanopipettes, e.g., Refs. �13,14�. Bysing boundary conditions that represent either a naturally or ar-ificially constrained membrane, we investigate the stress stateithin a patch of membrane when it is acted upon by a single

ctin filament or microtubule.Our results are also twofold: one scientific and one technologi-

al. The scientific point of interest is in investigating the limitsmposed by nature upon the relative mechanical strengths of theytoskeleton versus the cell membrane. For example, if the cellembrane were to have evolved as a structure that was incapable

f flexibility, this would not allow for cell motility. Flexibilitymplies a thin structure, which typically implies a weak structure.owever, if the cell membrane were too weak to sustain either

nvironmental or growth and movement loads, this would com-romise the cell’s integrity leading to a reduced fitness. Usingimilar reasoning, a cytoskeletal element such as actin or tubulinhat was too weak to maintain cell shape would result in an unvi-ble cell as it would be incapable of growth or movement at a rateufficient to compete with sister cells or other cells occupying aimilar ecological niche. If, on the other hand, the cytoskeletalctin or tubulin formed structures that were so strong as to rupturecell membrane, this would also lead to an unfit species.Results obtained through our computer simulations can be used

o not only better understand the force balance necessary for cellrowth, motility, and mechanotransduction but also to determinehe forces and stresses required to puncture a biological mem-rane. This has potential for single cell injection and in vitro fer-ilization applications.

ethod of Approach

Model Parameters. In the present model, we assume that theembrane is homogenous, ignoring the effect of inhomogeneities

aused by membrane proteins. We also begin with the assumptionhat under quasi-static loading conditions, the membrane may be

odeled as a transversely isotropic elastic material, with theembrane normal serving as the axis of symmetry. A transversely

sotropic material requires that five elastic constants be specified.ince the cell membrane has been observed to behave as a two-imensional fluid with the two layers of the membrane flowingmoothly past each other and with individual phospholipid mol-cules diffusing freely and exchanging locations �17,18�, we havehosen to model both in-plane and out-of-plane Poisson’s ratios as.49. Indeed Charras et al. �19� previously assumed Poisson’s ra-
io of 0.49 for the cell membrane. This ratio was based on a paper
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by Hamill and Martinac �20� in which the cell membrane is as-sumed to be incompressible. To our knowledge, this is yet to beexperimentally verified or theoretically predicted. In-planeYoung’s moduli, Yr� were set equal to 128 MPa. This was com-puted by dividing the area expansion modulus of a lecithin:cho-lesterol �SOPC:CHOL� lipid vesicle, which is representative of aneuronal growth cone, by the thickness of the membrane,�4.5 nm �21–23�. The area expansion modulus of the membraneis related to the bending modulus, �, used by Atilgan et al. �12�through

� � KAt2/24 �1�

where KA is the area expansion modulus of the membrane and t isthe thickness of the membrane �24,25�. The bending modulus isthen related to Young’s modulus through

� � Yt3/24 �2�

Using Eq. �1�, the bending modulus calculated for the SOPC:CHOL membrane is 4.9�10−19 N m, greater than the bendingmodulus used by Atilgan et al. �12�, 8.6�10−20 N m. The greaterbending modulus of the SOPC:CHOL membrane is likely causedby its cholesterol content �24�. Note that Atilgan et al. �12� did notuse an explicit membrane thickness, thus discrepancies may alsolie in the third-order relationship between thickness and bendingmodulus. Circular patches of membrane with radii of 50 mm and250 nm, a range over which a single actin filament or microtubuleis expected to act, were modeled �15,26�.

For an isotropic material with Poisson’s ratio of 0.5, the shearmodulus is equal to Y /3, and therefore 43 MPa. This value waschosen for the through-plane shear moduli, Grz and G�z. However,because the layers of the membrane flow freely over one another,the in-plane shear modulus, Gr�, must be lower. This was taken tobe 1.28 MPa, two orders of magnitude smaller than the out-of-plane shear modulus, small enough to approximate a fluid, whilelarge enough to maintain numerical stability.

In addition to the membrane properties, a quantification of theamount of force that the cytoskeleton can exert against a mem-brane is needed to understand the stress state of the cell mem-brane. Generally this force, Fcrit, is taken to be the amount of forcethat a straight filament can withstand in compression before first-mode buckling,

Fcrit =EI�2

L2 �3�

where EI is the flexural rigidity, and L is the length of the cyto-skeletal filament.

The maximum deformation of the membrane will occur underthe maximum force prior to cytoskeletal buckling. Once the buck-ling force is reached, additional force will cause the filament tobend but with only minimal additional corresponding membranedeformation. Instead, a continued increase in force will result infilament deformation and eventually filament collapse �27�. Thisphenomenon has been demonstrated by aspirating liposomes con-taining microtubules �17�. Thus, to determine the maximum stressin the membrane, forces only up to the filament buckling areconsidered.

A transverse loading of the cytoskeletal element will also occurdue to the force generated as the membrane of the growth coneslides along the tip of the protein during membrane extension.This force is dependent on the viscosity of the membrane and theextension rate of the growth cone, via

F = �v �4�

where viscosity � is approximately 0.00021 dynes s /cm and ve-locity � is 0.003–0.25 �m /s, resulting in a force of 0.0063–0.053 pN �18�, or approximately one to three orders of magnitudesmaller than the normal load on the cytoskeletal element. Theseforces are therefore not considered in our model. Indeed, Dai and
Sheetz �28� stated that the viscous forces are too small to cause a
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ignificant effect on growth rate and extension rates. To determinehe maximum stress in the membrane, only a force up to thelament buckling force is considered, since it is likely to be99.9% of the total force experienced by filament bending and

ompression �27�.In general, the axis of a cytoskeletal element will interact with

he surface of the membrane at an angle � �Fig. 1�.This is demonstrated in neuronal and mobile cells that contain

he protein complex Arp2/3, which causes actin filaments toranch from mother actin filaments at angles of 70 deg �2,29�.his results in a meshwork of cortical actin filaments acting at aide range of angles to the surface of the membrane. When therotein and membrane are not aligned normally, the magnitude oforce that can be exerted against the membrane before the proteineforms is

F = Fcrit sin � �5�

here � is the angle between the filament and the surface of theembrane.Once the amount of force that a filament can support was com-

uted, the normal contact stress exerted on the membrane due tohis force was calculated. This normal contact stress is P=F /A,here P is the normal contact stress of the filament acting on theembrane and A is the area of the protein that acts against theembrane. An actin filament was treated as a solid cylinder with

rea, Aa=�ra2, where ra is the radius of an actin filament, �4 nm

2,25�. The microtubule was taken as a hollow cylinder, and therea of the microtubule that acts against the membrane is AMT�rMTo

2−�rMTi2, where rMTo is the outer radius of the microtu-

ule, �12.5 nm, and rMTi is the inner radius of the microtubule,8.5 nm �2,25�.Reported values for the flexural rigidities, EI, of actin filaments

ange from 10−27 N m to 10−26 N m2 �10,15,16,30,31�. Two ex-remes of the values reported for actin flexural rigidity, 0.40

10−26 N m and 7.3�10−26 N m2 were used in our simulations7,21�. By contrast, the flexural rigidity of individual microtubuless �5.0�10−24 N m2 �32–34�, and for multiwalled carbon nano-ubes with a diameter of �50 nm, the flexural rigidity is �2.1

10−19 N m2 �22�.Since we are interested in the interaction of single filaments

ith the membrane, such as may be found in a neuron at thenception of axon or dendrite development or in an actively grow-ng lamellipodial or filopodial region where a new branch may beorming, we chose to simulate an actin filament with a length of00 nm. By contrast, aligned actin filaments within the lamellipo-ia and filopodia, which are actively contributing to filopodialrowth are �0.5–5 �m in length, and filaments found in theranched networks tend to be shorter �19,35,36�.

Using 0.40�10−26 and 7.3�10−26 N m2 for the actin flexural

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ig. 1 Scale cartoon drawing of an actin filament with a diam-ter of approximately 8 nm impinging upon the inner surface ofphospholipid bilayer membrane with a thickness of approxi-ately 4.5 nm.

igidities and 300 nm for the filament length, two different values

ournal of Biomechanical Engineering


for Fcrit were determined �Eq. �3��. Next, F was computed forfilament angles ranging from 10 deg to 90 deg, in increments of10 deg �Eq. �5��. This gave us a total of 18 forces, ranging from0.0763 pN to 8.01 pN. This range of forces corresponds to a mi-crotubule with a flexural rigidity of 5.0�10−24 N m2 and lengthbetween 2.5 �m and 25 �m and a multiwalled carbon nanotubewith a flexural rigidity of 2.1�10−19 N m2 and a length between509 �m and 5000 �m, both acting perpendicularly to the mem-brane �Eq. �3��.

The areas in which the protein filaments will act against themembrane for an actin filament and a microtubule are �50 nm2

and 263 nm2, respectively �2,25�. Given the calculated range offorces, the normal contact stresses exerted on the membrane bythe actin filament is 0.0015–0.1602 MPa. The normal contactstresses exerted by an actin filament acting on an �50 nm2 area inthe center of the membrane were modeled, and the deformationsand strains of the membrane were determined.

Boundary Conditions. Simulations with two different bound-ary conditions �BCs� were performed �Fig. 2�. For BC1, the edgeof the membrane was clamped and immovable: Dirichlet BCs atthe boundary, . This BC represents a bounding case that maybe likened to what a cell might encounter if were loaded veryquickly, such that the membrane had insufficient time to “flowinto the boundary,” as in the case of a cell injection. Indeed, suchpractice is used by modern cell injectors that use a very fast jab-bing or hammering motion to inject a neuronal growth cone oncethe pipette tip has been brought into close proximity �23� �per-sonal observation�. To our knowledge, no such boundary condi-tion has been observed in vivo, but might represent a conditionwherein the cell membrane was under osmotically induced tensionand unable to flow freely or if it were tightly constrained by inte-gral proteins interacting with both the cytoskeleton and the extra-cellular matrix, thereby restricting membrane flow. For BC2, weemployed Neumann–Dirichlet conditions, by constraining move-ment in the through-plane condition and by specifying the slope,w /r=0, at the boundary. Two additional boundary conditionswere supplied at the center of the membrane as a mixed BCwherein the slope w /r=0 due to symmetry, and the loadingforce was specified. The clamped-movable condition of BC2 isequivalent to that used by Atilgan et al. �15�.

Fig. 2 Schematic of a circular membrane deforming under apoint force acting at center of the membrane. The membrane isrepresented by �, and the edge of the membrane by ��. Polarcoordinates, r, z, � were used. „a… The top view and side view,respectively, of a clamped and immovable membrane along��, where �dw /dq��=0 and q=qo. „b… The top view and sideview, respectively, of a membrane clamped along �� but free tomove in q, �dw /dq��=0, but qÅqo. The dashed lines depict thedeformed membrane „not to scale….

Both our BC2 and that of Atilgan et al. �15� allow the mem-

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rane to flow freely in the radial direction. However, our BC2llows the boundary to move in as the membrane is deformed.C1, on the other hand, restricts the in-plane flow of the mem-rane, and therefore when a force acts on the membrane it willroduce greater areal strain. Without providing any restrictions,he membrane can essentially be pulled to infinity, as modeled bytilgan et al. �15�. In living cells, the flow and the growth of theembrane are restricted by integral proteins and their numerous

nteractions with both the cytoskeleton and the extracellular ma-rix. These interactions and inhomogeneities will thus affect thetress state in the membrane when a load is applied. These com-lexities, however, are reserved for future work.

Since the diameter-to-thickness ratio of our simulations rangedrom �20 to 200, a large deflection analysis was used to accountor the stresses in the middle plane of the membrane as the mem-rane deformed.

Numerical Versus Analytical Solution. We used the Timosh-nko membrane model �35� for validation of our numerical model.

comparison of the finite element analysis with the analyticalolution was performed to ensure that ANSYS is capable of accu-ately modeling the geometries and boundary conditions used inur model. The analytical solutions for our simulations were per-ormed for an isotropic membrane with a Poisson’s ratio of 0.3nd an elastic modulus of 128 MPa. For validation purposes, aoint force at the center of the membrane as opposed to a normalontact stress distribution was applied to the membranes. The ana-ytical solution includes the deformation and stresses caused by

embrane bending and stretching stresses. It utilizes the follow-ng equations to solve for the maximum deflection of the mem-rane, wo, bending stresses at in the r and � directions, �rr and��, respectively, and midplane tensile stresses at in the r anddirections, �rr� and �� , respectively,

wo

t+ A�wo

t3

= BFa2

Yt4 �6�

�rr = �rrYwot

a2 �7�

�� = ��Ywot

a2 �8�

�rr� = rrYwo

2

a2 �9�

�� = ��Ywo

2

a2 �10�

here t is the thickness of the membrane, F is the applied force, as the radius of the membrane, Y is the elastic modulus of the

embrane, and A, B, �rr, ��, rr, and �� are the constantserived from the approximate solution using the Galerkin equa-ions. In determining the most appropriate failure criteria for as-essing the stress value at which the membrane ruptures, we con-idered several stress formulations. Among those considerednclude the maximum principal stress, the areal strain, the Trescatress, and the von Mises stress. We decided upon the areal strain,efined the sum of the in-plane strains, since it is the only knownailure criterion previously considered for membrane rupture. Welso report the von Mises stress criterion for the following rea-ons: �1� the von Mises stress offers a means of comparing thetress states of any stress condition to that of tensile stress; �2� theresca stress is primarily for modeling plastic flow and lies within

he von Mises ellipse; �3� the von Mises stress represents thenergy per unit volume that a material may sustain prior to failure,nd it gives a field of the scalar equivalent of the stress state,hich in the case of our membrane is a compound loading con-
ition: bending and tension. There are also precedents for using
21006-4 / Vol. 131, FEBRUARY 2009


von Mises stress to quantify membrane stress. The von Misesstress, �e, at the edge of the membrane under plane stress condi-tions was determined from �rr and �� by

�e =��rr + �rr�� − �� + ��2 + ��rr + �rr��

2 + �� + ��2

2

�11�A comparison of the numerical solution and analytical solution

was made for the maximum membrane deflection and von Misesstress along the edge of the membrane. We compared the resultsof 54 total simulations: three different diameter membranes�100 nm, 200 nm, and 500 nm�, BC1 and BC2, and nine calcu-lated forces �0.076–8.01 pN�.

Element Type. Because the membrane was modeled as a cir-cular section, with symmetric boundary conditions, only axisym-metric element types were chosen for comparison. Atilgan et al.�15� also assumed symmetric boundary conditions. The use ofaxisymmetric elements allowed us to model only one section ofthe membrane but was representative of the entire membrane.This permitted us to use fewer elements in our simulation, result-ing in faster computation times. Preliminary simulations run with3D tetrahedral elements resulted in identical results to those foundwith the 2D axisymmetric. However, once we surpassed a radiusof 50 nm, the 32,000 node limit of our ANSYS license was ex-ceeded. Since the structure is very high aspect ratio, and since 3Delements must maintain an aspect ratio close to unity, adding moreelements of a different shape was not possible.

The membrane modeled in our simulations had a largediameter-to-thickness ratio, similar to a shell structure. Two dif-ferent shell element types, SHELL208 and SHELL209, as well asa 2D plane element type PLANE183, were used to model ourmembrane. SHELL208 and SHELL209 are typically used tomodel thin to moderately thick shell structures. SHELL208 ele-ments contain two nodes, and SHELL209 elements contain threenodes. The nodes in each of the shell types contain three degreesof freedom: two translational and one rotational. PLANE183 is aneight-node element, each node having two translational degrees offreedom.

The deformation resulting from the use of three element typeswere compared to the analytical solution, �Eq. �6�� for a clamped-immovable membrane with a radius of 250 nm. The von Misesstresses as calculated from Eq. �11� were also compared. Usingelement type PLANE183, deflection errors ranged from 0.11% to2.20% and errors in von Mises stress along ranged from26.49% to 35.45%. For these membrane dimensions and boundaryconditions, SHELL208 elements produced deflection errors rang-ing from 1.63% to 3.74% and stress errors along from 0.81%to 7.88%. The larger errors in von Mises stress made PLANE183an unfavorable element type.

Comparisons were then made between the two shell elementtypes and the analytical solution for a clamped-immovable mem-brane with a 250 nm radius and a force of 1.39 pN acting at themembrane center. This force was chosen because it is an interme-diate of the forces considered in our simulations. Deflection errorsat the center of the membrane and error in von Mises stress along were compared for the two shell types with varying number ofelements. Element types SHELL208 and SHELL209 yielded simi-lar results, with similar percent errors from the analytical solution�Tables 1 and 2�.

The trend for percent error in the deflection and von Misesstress calculations showed a decrease in error as the number ofelements increased. The percent errors for membranes composedof ten or more elements for both shell types were �1% of eachother. The element type SHELL208 was chosen because it hadfewer nodes and therefore a faster computational time thanSHELL209.

Convergence Study. A strain convergence study was per-



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ormed to determine the number of elements needed to model theembranes given the largest normal contact stress, 0.1602 MPa,

cting against the membrane. Preliminary studies demonstratedhat the larger the normal contact stress, the greater the number oflements needed for the maximum strain in any element of theembrane to converge. The convergence study was performed for

oth BC1 and BC2 on membranes with 50 nm and 250 nm radii.he maximum strain in the radial direction on the outer surface of

he membrane was determined numerically, and the results wereompared for membranes comprised of 100, 1000, and 2500 ele-ents for the 50 nm radii membranes and 500, 1000, and 10,000

lements for the 250 nm radii membranes. The 2500 element so-ution was used as the convergence criterion for the 50 nm radii

embranes, and the 10,000 element solution was used as the con-ergence criterion for the 250 nm radii membranes. Increasing theumber of elements in our simulation beyond these numbers re-ulted in high-aspect-ratio elements with radial dimensions ap-roaching zero.

The convergence study indicated that a 1000 element mesh wasufficient to model a membrane with a radius ranging from0 nm to 250 nm. Using 1000 elements, the membranes with ra-ii of 250 nm, under a normal contact stress of 0.1602 MPa con-erged within �2% of the 10,000 element solutions. For the0 nm radii membranes, the 1000 element membranes convergedo a solution within �3.5% of their 2500 element solution �Fig.�.

Deformations and Stresses in Neuronal Membranes. Aftererifying that ANSYS is capable of solving the simulation givenur geometries and boundary conditions, we computed the deflec-ions and stains of neuronal clamped-immovable membranes,C1, and clamped-movable membranes, BC2, of 50 nm and

able 1 Percent error of the numerical solution compared tohe analytical solution for the maximum deflectin of a clamped-mmovable 250 nm radius membrane with a force of 1.39 pNcting at its center

lement type No. of elements % error

HELL208 4 21.0110 7.3525 2.8050 2.15

100 1.50

HELL209 4 13.8610 6.7025 3.4550 2.15

100 1.50

able 2 Percent error of the numerical solution compared tohe analytical solution for the von Mises stress along �� for alamped-immovable 250 nm radius membrane with a force of.39 pN acting at its center

lement type No. of elements % error

HELL208 4 10.8210 4.1825 1.2050 0.17

100 0.34

HELL209 4 12.0910 4.3825 1.2350 0.17

100 0.38



250 nm radii. Neuronal membranes were modeled as transverselyisotropic with the values listed in Table 3.

ANSYS was also used to determine the areal strain in the mem-brane. If the areal strain in a single element of the membraneexceeded �3%, the membrane was considered to have ruptured�39,40�. The areal strain of an element is the sum of its in-planestrains such that

�areal = �rr + �� 12�

where �areal is the areal strain, �rr is the radial in-plane membranestrain, and �� is the circumferential in-plane strain. While thestrain energy contained in a single actin filament or microtubule isinsufficient to cause membrane rupture or puncture, this is not thecase with micropipettes or nanopipettes. Since our load was ap-plied on an area with a diameter of 8 nm, representative of anactin filament, it is noted that the forces necessary to cause a localareal expansion of 3% are likely to be lower than those imposedby larger structures such as those used in typical tethering experi-ments. In an experiment performed by Dai and Sheetz �18�, using0.5 �m microspheres, the tether force, or the force required toproduce and maintain a tether in a DRG growth cone, was deter-mined to be �6.7 pN. However, using much larger 4.0 �m mi-crospheres, another group found a steady-state tether pulling forceof 246.3 pN for human embryonic kidney cells �30�. Thus, fromthese two papers alone, we may conclude that the tether forcescalculated are dependent on the size of the microspheres andthereby the radius of membrane being pulled. In our simulations,the load is applied on an 8 nm diameter circular area, thus simu-lating a much greater strain gradient and resulting stress gradientwithin the membrane, leading to a smaller force to produce defor-mation and, under the right conditions, rupture or puncture.

In addition to simulation data, the normal contact stresses that abuckling �31� and a breaking �32� microtubule can exert on themembrane were calculated, and these values were included in oursimulations. The flexural rigidity of the microtubules used forthese calculations was estimated to be 5�10−24 N m2 �32–34�.The length of the breaking microtubule was determined visuallyfrom Ref. �32� to be 4 �m, and the length of the bending micro-tubule was listed in Ref. �31� as 20 �m.

Results

Numerical Versus Analytical Solutions. The percent error for

Fig. 3 Numerical solution for the maximum membrane strainfor clamped-immovable and clamped-movable membraneswith radii of 50 nm and 250 nm versus the number of elementsused to create the membranes. The clamped-immovable mem-branes are represented by circles, and the clamped-movablemembranes are represented by triangles.

deformation and maximum von Mises edge stress were computed

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or three different membrane sizes with both boundary conditions.comparison was made between the analytical solution and the

umerical solution for an isotropic membrane with Poisson’s ratiof 0.3, Young’s modulus of 128 MPa, and shear modulus of3 MPa, subjected to a point force. The clamped-immovableembrane, BC1 50 nm membrane, had an error in the deforma-

ion calculation of about 18% �Fig. 4� and an error in von Misesdge stress ranging from 0.00% to 1.49% �Fig. 5�. The membraneith a 100 nm radius had a deformation error ranging from 5.40%

o 7.86% and an edge stress error of 0.00–1.29%. The error for the50 nm radius membrane had a deformation error range of 1.63–.74% and an edge stress error ranging from 0.00% to 7.89%. Forhe 250 nm radius membrane, as the force increased, the errorended to increase as well.

When the edge of the membrane was movable, BC2, the percentrrors for the membrane deformation and edge stress were similarFigs. 6 and 7�. For these boundary conditions, the 50 nm mem-rane had a deflection error �18%, with an error in edge stressanging from 0.00% to 2.17%. The 100 nm membrane had a de-ection error from 5.40% to 7.86% with a stress error from 0.00%

o 2.17%. Finally, the 250 nm membrane had a deflection errorrom 1.89% to 3.52% and a stress error from 0.00% to 13.06%.or the 250 nm radius membrane, as the force increased, the error

ended to increase as well.

Deformations and Stresses in Neuronal Membranes. Theow percent error computed when comparing the analytical solu-ion to the numerical solution for an isotropic membrane demon-

Table 3 Mo

Parameter Description

a Membrane diameterAa Area in which actin

filament can expert forceEIa Flexural rigidity of actin filameF Filament force acting on membra

Gr� In-plane shear modulus of membrGzr and G�z Out of plane shear moduli of the me

P Normal contact stress acting of memra Radius of actin filamentt Thickness of neuronal membran

Yrr, Yzz, Y�� Young’s modulus of membrane� Angle in which filament acts

�r�, �zr, ��z Poisson’s ratio membrane

ig. 4 Maximum membrane deflection in the z direction ver-us applied force for clamped-immovable membranes of sizes0 nm, 100 nm, and 250 nm in radii with Poisson’s ratio of 0.3.he analytical solution is represented by the solid line and the
NSYS solution with the open circles.
21006-6 / Vol. 131, FEBRUARY 2009


strates that our model is accurate for the given forces and geom-etries. The membrane model was then changed to represent thetransversely isotropic neuronal membrane. A normal contact stressdistributed over the area in which an actin filament would act,50 nm2, was substituted for the point load. Since ANSYS does nothave a transversely isotropic material type, an orthotropic materialtype was chosen. Of the nine constants necessary to define anorthotropic material, five unique values were entered, as summa-rized in Table 3. The low in-plane shear modulus was selected tosimulate a fluid incapable of supporting shear load. Although thestatic in-plane shear modulus of the membrane is unknown, theability of the layers of the membrane to flow freely across oneanother indicates that the membrane has a low in-plane shearmodulus. The membrane strains and deformations were then re-ported for three surfaces of the membrane: inner �the inner surfaceof the membrane in which the filament acts�, outer �the surface ofthe membrane outside the cell�, and the midplane of the mem-brane. Membrane failure, or rupture, is expected to occur if themaximum areal strain in any surface of any element of the modelexceeded 3% �25�. The areal strain at any location in the mem-brane is equal to the sum of the in-plane strains for that particularelement per equation �12�.

Boundary Condition 1. The bending and breaking forces formicrotubules taken from the literature were modeled as exertingforces of 0.17 pN and 3.1 pN, respectively �Eq. �3��. The force in

parameters

Value Reference

100 nm, 200 nm, 500 nm �26��50 nm2 �2,25�

0.40�10−26, 7.3�10−26 N m2 �7,21�0.0763–8.01 pN �10,35,30�

1.28 MPaane 43 MPane 0.0015–0.1602 MPa �10,35,30�

4 nm �2,25�4.5 nm �25�

128 MPa10–90 deg —

0.49 �37,38�

Fig. 5 von Mises stress along �� versus applied force forclamped-immovable membranes of sizes 50 nm, 100 nm, and250 nm radii with Poisson’s ratio of 0.3. The analytical solutionis represented by the solid line and the ANSYS solution with theopen circles. The 50 nm and 100 nm stresses were nearly

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hich a polymerizing microtubule was measured to exert againstbarrier is within this range, �2.5 pN �33�. Forces of 0.17 pN

nd 3.1 pN correspond to normal contact stresses of 0.00065 MPand 0.012 MPa, respectively, given the geometry of the microtu-ule. The maximum normal contact stress the experimentally ob-erved microtubule can exert is within the values tested in ourimulations. For a 250 nm cell membrane with a clamped-mmovable edge, BC1, a normal contact stress of 0.1602 MPacting on an 8 nm diameter circular area at the center of the mem-rane deflected the membrane, 5.81 nm. The maximum arealtrain in the membrane under this force was �0.17% �Eq. �12��.his maximum strain occurred at the element in the center of thetress distribution on the outer surface of the membrane. To pre-ent rupturing of the membrane, the maximum normal contacttress that can be applied to a membrane under these boundaryonditions is �2.55 MPa, many times greater than that expectedo be exerted by the cytoskeleton but less than that imposed by a

ig. 6 Maximum membrane deflection in the z-direction ver-us applied force for clamped-movable membranes of sizes0 nm, 100 nm, and 250 nm radii with Poisson’s ratios of 0.3.he analytical solution is represented by the solid line and theNSYS solution with the open circles.

ig. 7 von Mises stress along �� versus applied force forlamped-movable membranes of sizes 50 nm, 100 nm, and50 nm radii with Poisson’s ratios of 0.3. The analytical solu-ion is represented by the solid line and the ANSYS solution withhe open circles. The 50 nm and 100 nm stresses are nearly
dentical.


glass micropipette or nanopipette, e.g., Ref. �13�. The maximumnormal contact stress before rupture was determined by dividingthe maximum allowable areal strain, 0.03, by the maximum strainin the membrane when 1 MPa of normal contact stress is applied�Fig. 8�. Our results predict that rupture will first occur at thecenter of the stress distribution on the outer surface of themembrane.

For the 50 nm membrane, a 0.1602 MPa normal contact stressdisplaced the membrane, 1.05 nm. The maximum areal strain inthe membrane given these conditions is �0.13%. Our model pre-dicts that rupture will first occur at the center of the stress distri-bution on its outer surface at a force of approximately 3.74 MPa�Fig. 9�.

Boundary Condition 2. A 250 nm membrane with a clamped-

Fig. 8 Areal strain „εareal… /MPa versus radial distance from thecenter of the membrane for a clamped-immovable membranewith a radius of 250 nm. Membrane rupture will occur when thetension in the membrane causes the areal strain to exceed 3%.This will occur at center of the membrane on its outer surface ifthe normal contact stress applied exceeds È2.55 MPa.

Fig. 9 Areal strain „εareal… /MPa versus radial distance from thecenter of the membrane for a clamped-immovable membranewith a radius of 50 nm. Membrane rupture will occur at centerof the stress distribution on its outer surface if the normal con-
tact stress applied exceeds È3.74 MPa.
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ovable edge, BC2, under a normal contact stress of 0.1602 MPaaused membrane deflection of 6.87 nm. The greatest areal strainn the membrane under this force is �0.17%. The maximum nor-

al contact stress that can be applied to this membrane withouthe membrane rupturing is �2.61 MPa. Rupture will first occur athe center of the normal contact stress distribution on its outerurface �Fig. 10�.

For the 50 nm membrane, a 0.1602 MPa normal contact stressisplaced the membrane, 1.08 nm. The maximum areal strain inhis membrane given the movable boundary conditions is

0.11%. The normal contact stress exerted on this membraneannot exceed �4.09 MPa without membrane rupture occurringn its outer surface at the center of the normal contact stressistribution, Fig. 11.

ig. 10 Areal strain „εareal… /MPa versus radial distance fromhe center of the membrane for a clamped-movable membraneith a radius of 250 nm. Membrane rupture will occur at theenter of the normal contact stress distribution on its outerurface if the normal contact stress applied exceeds2.61 MPa.

ig. 11 Areal strain „εareal… /MPa versus radial distance fromhe center of the membrane for a clamped-movable membraneith a radius of 50 nm. Membrane rupture will occur at the cen-

er of the normal contact stress distribution on its outer surface
f the normal contact stress applied exceeds È4.09 MPa.
21006-8 / Vol. 131, FEBRUARY 2009


ConclusionsWe modeled the stress distribution in a neuronal membrane

under localized contact force. This model represents the loadingconditions likely to occur under a variety of conditions. One is theinteraction between the cell membrane and a single actin filament,as in the case of lamellipodial extension. The second is the inter-action of the cell membrane with a single microtubule. The thirdloading case is one in which a nanopipette impinges upon the cellmembrane during an injection. We conclude similarly to Atilgan etal. �15� that a single actin filament with a length of �300 nm isincapable of deforming the membrane into a filopodial shape priorto its buckling. Furthermore, we conclude that the membranestresses likely to be generated by microtubules loaded axially incompression as they impinge upon a cell membrane as computedfrom the experimental data of Odde et al. �32� will not cause themembrane to exceed the areal strain failure criterion given forphospholipid bilayers by Boal �25�. The 0.00065 MPa normalcontact stress estimated from a bending microtubule and the0.0120 MPa normal contact stress estimated for the buckling mi-crotubule are too small to rupture a membrane ranging from50 nm 250 nm in radius under the constrained boundary condi-tions. Thus we predict that both a 100 nm diameter patch of mem-brane and a 500 nm diameter patch of membrane have “naturalmechanical safety factors” of �15–25. It has been shown that thelateral reinforcement of microtubules can allow them to sustainforces of �30–100 pN �27,34�. However, even at this force, themaximum normal contact stress on the membrane is only�0.380 MPa. This increase in normal contact stress would stillnot cause our membranes to rupture, but it would reduce oursafety factor to be within �7–10. The implication is that naturalselection phenomenon early in life’s history may have “found”this mechanical relationship between cytoskeleton and cell mem-brane.

An understanding of the magnitude of forces between cyto-skeletal proteins and membranes is necessary for basic scienceand for cell and tissue engineering applications. A cytoskeletonstructure that is too stiff, either by nature, disease, or human de-sign, could potentially rupture a cell membrane. If the supportstructure is too compliant, however, it will not be able to drive andmaintain the morphology of the cell. Additionally, a change in thedeformation of the cytoskeleton, which could occur due to achange in the mechanical properties of the cytoskeleton or cellmembrane, may lead to cell damage �41�.

As cytoskeletal proteins and membranes coevolved, and indeedcontinue to evolve, presumably an optimization did occur andcontinues to persist wherein the material strength of the cytoskel-etal filaments are sufficient to deform the cell membrane at a ratesufficient to generate cell growth but insufficient to cause mem-brane rupture. Likewise, the phospholipids responsible for main-taining cell integrity must remain compliant enough to allowtransmembrane proteins to populate the membrane with adequatefrequency and density to sustain cell metabolism, but must alsoremain strong enough to resist rupture from either externally orinternally generated forces. This line of investigation brings newpossibilities and imposes limits on what may be achievable withcellular and tissue engineering projects as well as artificial lifeefforts.

Additionally, this work is a precursor to quantifying the normalcontact stresses required to penetrate a cell membrane during cel-lular injection, single cell surgeries, and organelle injections.Knowledge of these normal contact stresses is necessary to ensurethat minimal damage is done to the cell during these procedures.Our BC2 represents a boundary condition identical to that used byAtilgan et al. �15�. On the other hand, our BC1 represents a casewhere the membrane is undergoing rapid loading whereby themembrane flow response time is inadequate to respond. Bothboundary conditions produced deflection and stress results withinthe same order of magnitude. While this is surprising considering
that the membrane is allowed to flow radially under our BC2


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onditions, the results are consistent with the Timoshenko solu-ion. However, further experimental verification with probes onhe same scale as actual cytoskeletal elements is warranted.

There are a number of technological applications that wouldenefit from a better understanding of the response of a membraneo an applied. Specifically, the amount of normal contact stressequired to puncture a membrane for drug injection or in vitroertilization can be determined using simulation �36,37�. Thisould enable application of a force from a microfluidics device oranofluidics device to permeate the membrane of a cell, whileinimizing the chance of cell or organelle injury, e.g., Refs.

13,14�. This knowledge would allow for the automation of aorce-controlled injection system to become feasible. Whenmplementing haptic feedback into an injection device, a feedbacktarget,” generated through simulation, would benefit device cali-ration.

Future work will include modeling an aspirated spherical cellnd then determining the normal contact stresses required to punc-ure the membrane. Additionally, we will attempt to make theurrent model dynamic by incorporating the polymerization of theytoskeletal filaments against the membrane. The amount that theytoskeletal filament can grow, and thereby the amount of forcehe filament will be able to exert on the membrane, will be depen-ent on the tautness, or the size of the undulations, of the mem-rane. The more taut the membrane, the less accessible the end ofhe cytoskeletal element is to an additional protein subunit �38�.

cknowledgmentThis work was supported by a NSF fellowship award, by

Nanotechnology Meets Neuroscience” Pennsylvania Departmentf Health 4100026196-240418 and by the Keck Foundationhrough the Keck Center for Attofluidics at Drexel University.

omenclatureA � constant for solving maximum deflection of

membraneAa � area of actin filament acting against membrane

AMT � area of microtubule acting against membranea � radius of membraneB � constant for solving maximum deflection of

membraneEI � flexural rigidity of filament

�areal � areal strain in the membrane�r � in-plane strain in the membrane in the radial

direction�� in-plane strain in the membrane in the theta

directionF � force acting on membrane

Fcrit � buckling force of filamentGr� � in-plane shear modulus of the membrane

Gzr and G�z � out of plane shear modulus of the membraneL � length of filamentP � normal contact stress exerted on membrane by

protein filamentq � radial distance from center of membrane to

edge before deformationqo � radial distance from center of membrane to

edge after deformationr � radius of protein filamentt � thickness of the membrane

wo � maximum deflection of membraneYrr, Yzz, Y�� Young’s moduli of membrane

z � direction of deflection of membrane�r � constant for solving bending stress in r direc-

tion of membrane�� constant for solving bending stress in � direc-

tion of membrane
� � angle in which filament is acting on membrane


� � tangential direction along �r�, �rz, ��z � Poisson’s ratios of the membrane

�r � stress in membrane in r direction�v � von Mises stress in membrane�� stress in membrane in � direction r � constant for solving mid-plane tensile stress in

r direction of membrane � � constant for solving mid-plane tensile stress in

� direction of membrane � edge of membrane

� membrane

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Cytoskeleton-Membrane Interactions in Neuronal Growth Cones: A Finite Analysis Study